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Problem 79

# Let T: $$\mathrm{R}^{3} \rightarrow \mathrm{R}^{2}$$ be given by $\mathrm{T}\left(\mathrm{x}_{1}, \mathrm{x}_{2}, \mathrm{x}_{3}\right)=\left(7 \mathrm{x}_{1}+2 \mathrm{x}_{2}-3 \mathrm{x}_{3}, \mathrm{x}_{2}\right)$ As bases for $$\mathrm{R}^{3}$$ and $$\mathrm{R}^{2}$$ respectively, let $\mathrm{G}=\left\\{\mathrm{g}_{1}, \mathrm{~g}_{2}, \mathrm{~g}_{3}\right\\}=\\{(1,0,0),(0,1,-1),(0,0,1)\\}$ $\mathrm{H}=\left\\{\mathrm{h}_{1}, \mathrm{~h}_{2}\right\\}=\\{(1,0),(0,-1)\\}$

Expert verified
The matrix representation of the given linear transformation T with respect to bases G and H is: $T_G^H = \begin{pmatrix} 7 & 5 & -3 \\ 0 & -1 & 0 \end{pmatrix}$
See the step by step solution

## Step 1: Review the basics of the transformation

Given linear transformation T, which maps from R^3 to R^2: $T(x_1, x_2, x_3) = (7x_1 + 2x_2 - 3x_3, x_2)$ The bases for R^3 and R^2 are given by: $G = \{g_1, g_2, g_3\} = \{(1,0,0),(0,1,-1),(0,0,1)\}$ $H = \{h_1, h_2\} = \{(1,0),(0,-1)\}$

## Step 2: Evaluate the transformation for basis G

To determine the matrix representation of T with respect to bases G and H, apply T to each vector of basis G: $T(g_1) = T(1,0,0) = (7, 0)$ $T(g_2) = T(0,1,-1) = (5, 1)$ $T(g_3) = T(0,0,1) = (-3, 0)$

## Step 3: Express the transformed vectors in terms of basis H

Now, we need to express each transformed vector in terms of basis H: For $$T(g_1) = (7, 0)$$, $7h_1 + 0h_2 = (7, 0)$ For $$T(g_2) = (5, 1)$$, $5h_1 - 1h_2 = (5, -1)$ For $$T(g_3) = (-3, 0)$$, $-3h_1 + 0h_2 = (-3, 0)$

## Step 4: Create the matrix representation of T

From the coefficients obtained in Step 3, we can create the matrix representation of T with respect to bases G and H: $T_G^H = \begin{pmatrix} 7 & 5 & -3 \\ 0 & -1 & 0 \end{pmatrix}$ The matrix representation of the given linear transformation T with respect to bases G and H is: $T_G^H = \begin{pmatrix} 7 & 5 & -3 \\ 0 & -1 & 0 \end{pmatrix}$

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